Thursday, February 21, 2008

This is a version of http://www.southern-style.com/connected_math.htmCONNECTED MATHRequest for InformationBetty Peters, member of the Alabama State School Board, needs information on Connected Math (published by Pearson) which has been used in Baltimore, Maryland. The National Science Foundation claims it is "research-based, best practices." This would affect a large percentage of the schools in Betty's district. The information is critical. Please send your information to Betty Peters, bettyp@ala.net and Sharman Ramsey style@southern-style.com.

Response:I sent the request to two outstanding math groups that will fill in all the info you will need. Jimmy KilpatrickSenior Fellow, Alexis de Tocqueville Institutionhttp://www.adti.net Editor & Chief, EducationNews.orghttp://www.EducationNews.orgFrom MC webpage:

ANYWAY, this commentary is just a funny way of criticizing the whole issue...if you can tolerate reading the whole thing.

Tammy BrantleyConnected Math: A New PerspectiveI received this e-mail commentary recently, and I thought it was a riot. Hope you enjoy it, too! Tim

Some guys must be really really, like, hostile towards thedistrict to be making a federal case -- for real -- out ofthis whole big new math thing, aren't they? I wanted totry and share my feelings with you.I really really like and feel good about what I've heardabout the new math program, you know? I wish I'd hadsomething like it was I was a kid, because, you know, Iwas never any good at the traditional math and stuff. AndI feel like this new stuff is going to help kids like mewho were more, like, words and picture-oriented, and not so-- well -- like, linear thinkers. I mean, like, when I wasa kid, there were just a few kids who were really, like,focused on math. And everybody's different, you know, sothat's like okay for them, but it was tough on the rest ofus. And with diversity and all, everybody's got sort ofdifferent kinds of intellegence, and everybody's gottheir own personal learning style, and this new new mathstyle thing feels really sort of holistic and creative.Really, you know, it's like right-brain intellegence --sortof artistic and all, and there's maybe going to be kids getthis program that maybe never, you know, "got" real seriousmath before. And so, that's like a good thing for kids whowere like me.On the other hand, I'm feeling so much, like, hostility andnegativity here, and I feel like, you know, how when Ididn't get the old math, 'cause I was so creative and all,there're people -- parents I mean -- who just don't "get"the new math. And I've been meditating on that, and I feellike that anger grows out of their own learning styles, andthe kinds of intelligence they have. You know? It's like,I want to say to those guys, you guys ARE the linearpeople, the ones with really focused memories, and the oldmath really reached out for you, and you, like, embracedthe math and made it all part of yourselves. I mean -- andI still know people like this -- there are people who LIKEknowing how long division works or how to, I don't know,but say, turn some dumb fraction into a percentage. Thatwould be, for sure, such absolute torture for me -- but Ireally do know people who enjoy that sort of thing. They'rethe left-brained people, you know, and they have thatreally traditional linear thinking thing deep in theirinner-being. And maybe they aren't really creative -- ormaybe they are, I mean, or you are ... I shouldn't say.It's not like one kind of intelligence is better thananother -- so it's not like a DIS - ability --we're noteven supposed to use the term "learning disabled" anymore,right? -- so you parents are like, "Differently-Abled"because you're all linear and left-brained and so, like,strictly logical and just don't get the holistic creativediscovery math. But that's not wrong, I'm saying. It'sjust a different intellegence. And I keep telling myself Ihave to reach out, you know, and embrace the diversity.But I try to do that, and I feel all kinds of fear, and itfeels like it's really a valid fear. So maybe being linearpeople you all have, you know, linear kids. I'm not sayinglike anybody is born that way, -- or maybe they are... I'mnot a teacher, I don't know whether, you know, Nature /Nuture --whatever, but I _DO_ feel for sure that kids arereally really like their parents, right? And so if youparents are differently-abled and can't get the new math... I really am afraid the kids won't get it either.And when I was a kid, and I didn't get it, that was likeabout the worst, ever. And nobody made any allowances forme, because, back then you know _I_ was the one who wasdifferently-abled, only they treated me like I was trulystupid -- just a ditz, you know. But I'm not a ditz, I'mjust really creative. So I understand this new math thingyou're doing. But if a kid is handicapped by not being ascreative as I am, he may never figure it out. And that'swhat this takes, you know. Discovery learning.That's so cool if you're a discoverer, but linear peopleare so, like impatient and focused, they don't want to stopand figure it all out, they want to go straight to one bestanswer and move on. And so, a kid who grows up like that,and gets stuck in a discovery class where he's not, youknow, developmentally mature enough for holistic self-directed sorts of learning agendas, he (well, or maybe she)he's going to get all lost and bored and maybe be, youknow, disruptive or uncooperative. And I wonder if that'spart of what happened in, you know, Littleton, was kids whocouldn't "get it" and nobody reached out to them. That'sjust so sad, and makes me feel so awful, you know, I wantto reach _all_ the kids even those left-brainers. 'Cause akid who's differently-abled. might start behaving, youknow, inappropriately. And that would be bad for MY kidsor any kids who are trying to discover things. Because,you know, like this is a real team-think kind of processand if somebody is differently-abled on that team and theycan't participate in the discovery process, like --well,then, NOBODY is going to get anywhere. I mean, like,teachers can't be expected to teach one class with tworadically diverse kinds of learners. So linear kids who areso outside the discovery box, you know, -- especially ifthe parents are differently-abled in the same way as thekids -- they could like ruin the class for the others.It'd be like, trying to teach algebra in Vietnamese.Nothing says Vietnamese kids can't learn, but is it fair tomake the teachers work harder to reach them? I mean, whenthe parents and the whole background culture and thelearning styles and ---well, all that stuff is just more,well, sort of-- sort of repressed and focused, but it'snot _wrong_, really, but it's not fair to the teachers whohave to teach discovery learning to kids who really respondbest to guided or --what do they call it-- traditionalinstruction? And if a kid is left-brained, and isn'tgetting direct instruction like he needs, it's not fair tothose kids. And it's not fair for the linear minded_teachers_, you know. And I mean, I had a few of _those_;and I bet PISD still has some of those same ones. They'rethe sort who feel good about the old math, too, and I beta really creative administration could figure out how to,you know, match the creative teachers with the holistickids and the linear-minded teachers with that kind ofdifferently-abled kids. And that could make everybodyfeel good, which is, you know, really important for ourself-esteem, right?So like, even though the new new math is a good thing forcreative people like me, I feel like the way to maximizethe potentials of all the diverse kids, with many kinds ofintellegences and all, the schools still need to offer theold math as well as the new math.

California has had its share of educational crises - such as whole language and fuzzy math. Despite recent improvements, the state is still in the grips of an algebra crisis.The problem became apparent 20 years ago when the report "A Nation at Risk" warned of a "rising tide of mediocrity" in the public schools. The report claimed that too few students were taking the more rigorous courses in high school. Twenty years later, enrollment in college-prep courses is way up. Unfortunately, evidence indicates that student learning is about the same as it was back then.Importance of algebraRecent reports have stressed the importance of algebra in middle school; students who succeed in algebra usually do better in the rest of school and in their careers than those who do not. Well-intentioned school administrators often hope that early enrollment in algebra will reduce the achievement gap attributed to race or family income. Hence enrollments in middle-school courses called "Algebra" have increased. But judging from results on objective statewide tests, many middle-school students are not learning the subject, even those with passing grades.The strongest predictor of failure to learn algebra is not race or income; it is a lack of adequate academic preparation. The problem begins before students get to their first algebra class. Many school districts have watered down the content of pre-algebra courses, removing important but difficult material. The districts want more students to pass math classes, and they want to guarantee high pass rates by making the classes easy. But classes without content set students up for later failure in algebra.The depth of the problem varies. In some schools, the percentage of eighth-grade algebra students is moderately correlated to scores on the seventh-grade California Standards Test. In those schools, algebra readiness is still being used as part of the placement decision. In other schools, placement decisions appear unrelated to academic preparation. In the worst cases, all or nearly all students are placed in algebra by eighth grade, regardless of readiness.No district in California is more guilty of misguided placement strategies than the San Diego City Schools. The results are disastrous. Failing to learn algebra in eighth grade results in large numbers of students repeating algebra in ninth grade, even though success is not ensured the second time around.Admirably, California embraces learning algebra by the end of eighth grade as a long-term goal. But strengthening academics from kindergarten on is necessary before this goal can fully be met. Algebra placement rates ought to depend on student readiness.Seventh-grade student scores on the California Standards Test should guide placement in eighth-grade courses.State policyAnother state policy adds to the problem. As of now middle schools receive more credit on California's accountability index for eighth graders who take the algebra test than for those who take the general math test, encouraging schools to place too many students in eighth-grade algebra. The state should discourage that by taking away some credit on the accountability index for algebra exam failures.California's algebra crisis is serious but not terminal. Schools need to concentrate on improving students' readiness for algebra courses. Algebra for all is good, but without changes we could end up with algebra for none.Paul Clopton is a research statistician for the U.S. Department of Veterans Affairs in San Diego. Bill Evers is a research fellow at the Hoover Institution, for which this article was written.Paul Clopton is a research statistician for the U.S. Department of Veterans Affairs in San Diego. Bill Evers is a research fellow at the Hoover Institution, for which this article was written.

-

AN OPEN LETTER TO UNITED STATES SECRETARY OF EDUCATION, RICHARD RILEY

Dear Secretary Riley:In early October of 1999, the United States Department of Education endorsed ten K-12 mathematics programs by describing them as "exemplary" or "promising." There are five programs in each category. The "exemplary" programs announced by the Department of Education are:

Everyday MathematicsMathLandMiddle-school Mathematics through Applications Project (MMAP)Number PowerThe University of Chicago School Mathematics Project (UCSMP)These mathematics programs are listed and described on the government web site: http://www.enc.org/ed/exemplary/The Expert Panel that made the final decisions did not include active research mathematicians. Expert Panel members originally included former NSF Assistant Director, Luther Williams, and former President of the National Council of Teachers of Mathematics, Jack Price. A list of current Expert Panel members is given at: http://www.ed.gov/offices/OERI/ORAD/KAD/expert_panel/mathmemb.htmlIt is not likely that the mainstream views of practicing mathematicians and scientists were shared by those who designed the criteria for selection of "exemplary" and "promising" mathematics curricula. For example, the strong views about arithmetic algorithms expressed by one of the Expert Panel members, Steven Leinwand, are not widely held within the mathematics and scientific communities. In an article entitled, "It's Time To Abandon Computational Algorithms," published February 9, 1994, in Education Week on the Web, he wrote:

"It's time to recognize that, for many students, real mathematical power, on the one hand, and facility with multidigit, pencil-and-paper computational algorithms, on the other, are mutually exclusive. In fact, it's time to acknowledge that continuing to teach these skills to our students is not only unnecessary, but counterproductive and downright dangerous." (http://www.edweek.org/ew/1994/20lein.h13)In sharp contrast, a committee of the American Mathematical Society (AMS), formed for the purpose of representing the views of the AMS to the National Council of Teachers of Mathematics, published a report which stressed the mathematical significance of the arithmetic algorithms, as well as addressing other mathematical issues. This report, published in the February 1998 issue of the Notices of the American Mathematical Society, includes the statement:

"We would like to emphasize that the standard algorithms of arithmetic are more than just 'ways to get the answer' -- that is, they have theoretical as well as practical significance. For one thing, all the algorithms of arithmetic are preparatory for algebra, since there are (again, not by accident, but by virtue of the construction of the decimal system) strong analogies between arithmetic of ordinary numbers and arithmetic of polynomials."Even before the endorsements by the Department of Education were announced, mathematicians and scientists from leading universities had already expressed opposition to several of the programs listed above and had pointed out serious mathematical shortcomings in them. The following criticisms, while not exhaustive, illustrate the level of opposition to the Department of Education's recommended mathematics programs by respected scholars:

Richard Askey, John Bascom Professor of Mathematics at the University of Wisconsin at Madison and a member of the National Academy of Sciences, pointed out in his paper, "Good Intentions are not Enough" that the grade 6-8 mathematics curriculum Connected Mathematics Program entirely omits the important topic of division of fractions. Professor Askey's paper was presented at the "Conference on Curriculum Wars: Alternative Approaches to Reading and Mathematics" held at Harvard University October 21 and 22, 1999. His paper also identifies other serious mathematical deficiencies of CMP.R. James Milgram, professor of mathematics at Stanford University, is the author of "An Evaluation of CMP," "A Preliminary Analysis of SAT-I Mathematics Data for IMP Schools in California," and "Outcomes Analysis for Core Plus Students at Andover High School: One Year Later." This latter paper is based on a statistical survey undertaken by Gregory Bachelis, professor of mathematics at Wayne State University. Each of these papers identifies serious shortcomings in the mathematics programs: CMP, Core-Plus, and IMP. Professor Milgram's papers are posted at: ftp://math.stanford.edu/pub/papers/milgram/Martin Scharlemann, while chairman of the Department of Mathematics at the University of California at Santa Barbara, wrote an open letter deeply critical of the K-6 curriculum MathLand, identified as "promising" by the U. S. Department of Education. In his letter, Professor Scharlemann explains that the standard multiplication algorithm for numbers is not explained in MathLand. Specifically he states, "Astonishing but true -- MathLand does not even mention to its students the standard method of doing multiplication." The letter is posted at: http://mathematicallycorrect.com/ml1.htmBetty Tsang, research physicist at Michigan State University, has posted detailed criticisms of the Connected Mathematics Project on her web site at: http://www.nscl.msu.edu/~tsang/CMP/cmp.htmlHung-Hsi Wu, professor of mathematics at the University of California at Berkeley, has written a general critique of these recent curricula ("The mathematics education reform: Why you should be concerned and what you can do", American Mathematical Monthly 104(1997), 946-954) and a detailed review of one of the "exemplary" curricula, IMP ("Review of Interactive Mathematics Program (IMP) at Berkeley High School", http://www.math.berkeley.edu/~wu). He is concerned about the general lack of careful attention to mathematical substance in the newer offerings. While we do not necessarily agree with each of the criticisms of the programs described above, given the serious nature of these criticisms by credible scholars, we believe that it is premature for the United States Government to recommend these ten mathematics programs to schools throughout the nation. We respectfully urge you to withdraw the entire list of "exemplary" and "promising" mathematics curricula, for further consideration, and to announce that withdrawal to the public. We further urge you to include well-respected mathematicians in any future evaluation of mathematics curricula conducted by the U.S. Department of Education. Until such a review has been made, we recommend that school districts not take the words "exemplary" and "promising" in their dictionary meanings, and exercise caution in choosing mathematics programs.Sincerely,David KleinProfessor of MathematicsCalifornia State University, NorthridgeRichard AskeyJohn Bascom Professor of MathematicsUniversity of Wisconsin at MadisonR. James MilgramProfessor of MathematicsStanford UniversityHung-Hsi WuProfessor of MathematicsUniversity of California, BerkeleyMartin ScharlemannProfessor of MathematicsUniversity of California, Santa BarbaraProfessor Betty TsangNational Superconducting Cyclotron LaboratoryMichigan State UniversityThe following endorsements are listed in alphabetical order.William W. AdamsProfessor of MathematicsUniversity of Maryland, College ParkAlejandro AdemProfessor & ChairDepartment of MathematicsUniversity of Wisconsin-MadisonMax K. AgostonAssociate ProfessorDepartment of Mathematics and Computer ScienceSan Jose State UniversityHenry L. AlderProfessor of MathematicsUniversity of California, DavisFormer member of the California Board of EducationFormer President of the Mathematical Association of AmericaKenneth AlexanderProfessor of MathematicsUniversity of Southern CaliforniaFrank B. AllenProfessor of Mathematics Emeritus, Elmhurst CollegeFormer President, National Council of Teachers of MathematicsGeorge E. AndrewsEvan Pugh Professor of MathematicsPennsylvania State UniversityGregory F. BachelisProfessor of MathematicsWayne State UniversityMichael BeesonProfessor of Mathematics and Computer ScienceSan Jose State UniversityGeorge BiriukProfessor of MathmaticsCalifornia State University, NorthridgeWayne BishopProfessor of MathematicsCalifornia State University, Los AngelesGary J. BlanchardProfessor of ChemistryMichigan State UniversityCharles C. Blatchley, ChairDepartment of PhysicsPittsburg State UniversityMichael N. BleicherProfessor Emeritus,University of Wisconsin - MadisonChair, Department of Mathematical SciencesClark Atlanta UniversityJohn C. BowmanVice-PresidentNational Association of Professional EducatorsKhristo N. BoyadzhievProfessor of MathematicsOhio Northern UniversityBart BradenProfessor of MathematicsNorthern Kentucky UniversityStephen BreenAssociate ProfessorDepartment of MathematicsCalifornia State University, NorthridgeDavid A. BuchsbaumProfessor of Mathematics, EmeritusBrandeis UniversityFrank BurkProfessor of MathematicsCalifornia State University, ChicoAna Cristina CadavidProfessor of PhysicsCalifornia State University, NorthridgeGunnar CarlssonProfessor of MathematicsStanford UniversityDouglas CarnineProfessor of EducationUniversity of OregonDirector of the National Center to Improve the Tools of EducatorsMei-Chu ChangProfessor of MathematicsUniversity of California, RiversideSun-Yung Alice ChangProfessor of MathematicsPrinceton University and UCLAJeff CheegerProfessor of MathematicsCourant Institute, NYUOrin CheinProfessor of MathematicsTemple UniversitySteven ChuTheodore and Francis Geballe Professor of Physics and Applied PhysicsChair of PhysicsStanford University1997 Nobel Prize for PhysicsFredrick CohenProfessor of MathematicsUniversity of RochesterMarshall M. CohenProfessor, MathematicsCornell UniversityPaul CohenProfessor of MathematicsStanford UniversityRalph CohenProfessor of MathematicsStanford UniversityPeter CollasProfessor of PhysicsCalifornia State University, NorthridgeBruce ConradAssociate Dean College of Science and TechnologyTemple UniversityDaryl CooperProfessor of MathematicsUniversity of California, Santa BarbaraRobert M. CostrellDirector of Research and DevelopmentExecutive Office for Administration and FinanceCommonwealth of MassachusettsProfessor of EconomicsUniversity of Massachusetts at AmherstGeorge K. Cunningham, ProfessorDepartment of Educational and Counseling PsychologyUniversity of LouisvilleJerome DancisAssociate Professor of MathematicsUniversity of MarylandPawel DanielewiczProfessor, Department of Physics and AstronomyMichigan State UniversityErnest DavisAssociate Professor of Computer ScienceNew York UniversityMartin DavisProfessor Emeritus of Mathematics and Computer ScienceCourant InstituteNew York UniversityJane M. DayProfessor of Mathematics and Computer ScienceSan Jose State UniversityCarl de BoorProfessor of Mathematics and Computer SciencesUniversity of Wisconsin-MadisonPercy DeiftProfessor of MathematicsCourant InstituteNew York UniversityJohn de PillisProfessor of MathematicsUniversity of California, RiversideRobert DewarProfessor of Computer ScienceCourant Institute of Mathematical SciencesFormer Chair of Computer ScienceFormer Associate Director of the Courant InstituteNew York UniversityJim DoleProfessor and Chair of BiologyCalifornia State University, NorthridgeJosef DorfmeisterProfessor of MathematicsUniversity of KansasBruce T. DraineProfessor of Astrophysical SciencesPrinceton UniversityBruce K. DriverProfessor of MathematicsUniversity of California, San DiegoVladimir DrobotProfessorDepartment of Mathematics and Computer ScienceSan Jose State UniversityWilliam DukeProfessor of MathematicsRutgers UniversityJohn R. DurbinProfessor of MathematicsSecretary of the General FacultyThe University of Texas at AustinPeter DurenProfessor of MathematicsUniversity of MichiganMark DykmanProfessor of PhysicsMichigan State UniversityAllan L. EdelsonProfessor of Mathematics andVice Chair for Graduate AffairsDepartment of MathematicsUniversity of California, DavisYakov EliashbergProfessor of MathematicsStanford UniversityRichard H. Escobales, Jr.Professor of MathematicsCanisius College, Buffalo, NYLawrence C. EvansProfessor of MathematicsUniversity of California, BerkeleyBill EversResearch FellowHoover InstitutionStanford UniversityCalifornia State Academic Standards CommissionBarry FaginProfessor of Computer ScienceUS Air Force AcademyGeorge FarkasProfessor of PsychologyDirector, Center for Education and Social PolicyUniversity of Texas at DallasEditor, Rose Monograph Series of the American Sociological AssociationRobert FeffermanLouis Block Professor of MathematicsChairman, Mathematics DepartmentUniversity of ChicagoChester E. Finn, Jr.John M. Olin FellowManhattan InstituteFormer U.S. Assistant Secretary of EducationRonald FintushelUniversity Distinguished Professor of MathematicsMichigan State UniversityMichael E. FisherDistinguished Univeristy Professor & USM Regents ProfessorInsitute of Physical Sciences and TechnologyUniversity of MarylandWolf Prize in Physics, 1980Patrick M. FitzpatrickProfessor and ChairDepartment of MathematicsUniversity of MarylandYuval FlickerProfessor of MathematicsThe Ohio State UniversityGerald FollandProfessor of MathematicsUniversity of Washington, SeattleDaniel S. FreedProfessor of MathematicsUniversity of Texas at AustinDmitry FuchsProfessorDepartment of MathematicsUniversity of California, DavisDavid C. GearyProfessor of PsychologyUniversity of MissouriSamuel GitlerProfessor of MathematicsUniversity of RochesterSheldon Lee GlashowHiggins Professor of PhysicsHarvard University1979 Nobel Prize in PhysicsSimon M. GobersteinProfessor of MathematicsCalifornia State University, ChicoSteve GonekProfessor of MathematicsUniversity of RochesterJeremy GoodmanDepartment of Astrophysical SciencesPrinceton UniversityCo-founder, Princeton Charter SchoolJonathan GoodmanProfessor of MathematicsCourant Institute of Mathematical SciencesNew York UniversityDavid GossProfessor of MathematicsThe Ohio State UniversitySteven R. GossChairman of the BoardArizona Scholarship FundMechanical Engineer - Raytheon SystemsChristopher M. GouldProfessor of PhysicsDepartment of Physics and AstronomyUniversity of Southern CaliforniaMark L. GreenProfessor of MathematicsUniversity of California at Los AngelesBenedict H. GrossLeverett Professor of MathematicsHarvard UniversityLeonard GrossProfessor of MathematicsCornell UniversityPaul R. GrossUniversity Professor of Life Sciences (emeritus)University of VirginiaDina Gutkowicz-KrusinPrincipal ScientistElectro-Optical Sciences, Inc.Irvington, New YorkKamel HaddadAssociate Professor of MathematicsCalifornia State University, BakersfieldDeborah Tepper HaimoVisiting ScholarUniversity of California, San DiegoTrustee of Association of Members of the Institute for Advanced Study at PrincetonFormer President of the Mathematical Association of AmericaJoel HassProfessor of MathematicsUniversity of California, DavisDavid F. HayesProfessor of Mathematics and Computer ScienceSan Jose State UniversityDr. Adrian D. HerzogChairman, Deprtment of Physics and AstronomyCalifornia State University, NorthridgeMember Content Review Panel for California Science MaterialsRichard O. HillProfessor of MathematicsMichigan State UniversityE. D. Hirsch, Jr.University Professor of Education and HumanitiesUniversity of VirginiaDr. Hanna J. HoffmanSenior Laser ScientistIRVision, Inc.San Jose, CaliforniaDouglas L. InmanResearch Professor of OceanographyScripps Institution of OceanographyUniversity of California, San DiegoGeorge JenningsProfessor of MathematicsCalifornia State University, Dominguez HillsSvetlana JitomirskayaAssociate Professor of MathematicsUniversity of California, IrvinePeter W. JonesProfessor and Chair of MathematicsYale UniversityVaughan JonesProfessor of MathematicsMathematics DepartmentUC BerkeleyPeter J. KahnProfessor of Mathematics andSenior Associate DeanCollege of Arts and SciencesCornell UniversitySheldon KamiennyProfessor of MathematicsUniversity of Southern CaliforniaIlya KapovichAssistant Professor of MathematicsRutgers, The State University of New JerseyHidefumi KatsuuraProfessor of MathematicsSan Jose State UniversityJerry KazdanProfessor of MathematicsUniverity of PennsylvaniaDavid KazhdanProfessor of MathematicsHarvard UniversityLisa Graham KeeganSuperintendent of Public EducationState of ArizonaSharad KenyProfessor of MathematicsDepartment of MathematicsWhittier CollegeSteve KerckhoffProfessor of MathematicsStanford UniversityRobion C. KirbyProfessor of MathematicsUniversity of California at BerkeleySteven G. KrantzChairman and ProfessorDepartment of MathematicsWashington University in St. LouisSt. Louis, MissouriSergiu KlainermanProfessor of MathematicsPrinceton UniversityAbel KleinProfessor of MathematicsUniversity of California, IrvineKurt KreithProfessor Emeritus of MathematicsUniversity of California at DavisBoris A. KushnerProfessor of MathematicsUniversity of Pittsburgh at JohnstownTsit-Yuen LamProfessor of MathematicsUniversity of California at BerkeleySerge LangProfessor of MathematicsYale UniversityBenedict LeimkuhlerAssociate Professor of MathematicsUniversity of Kansasand Fellow, Kansas Center for Advanced Scientific ComputingNorman LevittProfessor of MathematicsRutgers University, New BrunswickJun LiAssociate Professor of MathematicsStanford UniversityPeter LiProfessor and Chair of MathematicsUniversity of California, IrvineAlexander LichtmanProfessor of MathematicsUniversity of Wisconsin-ParksideSeymour LipschutzProfessor of MathematicsTemple UniversityMei-Ling LiuProfessor of Computer ScienceCalifornia Polytechnic State UniversityDarren LongProfessor of MathematicsUniversity of California, Santa BarbaraJohn LottProfessor of MathematicsUniversity of Michigan - Ann ArborTom LovelessDirector, Brown Center on Education PolicyThe Brookings InstitutionWashington, DCSteve P. LundProfessor of GeophysicsDepartment of Earth SciencesUniversity of Southern CaliforniaWilliam G. LynchProfessor, Department of PhysicsMichigan State UniversityMichael G. LyonsConsulting Assoc. ProfManagement Science and EngineeringStanford UniversitySaunders Mac LaneMax Mason Distinguished Service Professor, EmeritusUniversity of ChicagoNational Medal of Science, 1989Former Vice President, National Academy of Sciences, 1973-1981Former Member, National Science Board, 1973-1979Michael MallerAssociate Professor of MathematicsQueens College of CUNYIgor MalyshevProfessor of MathematicsSan Jose State UniversityEdward MatzdorffProfessor of MathematicsCalifornia State University, ChicoMichael MayCo-Director, Center for International Security and Arms Control(Research) ProfessorDepartment of Engineering-Economic Systems and Operations ResearchStanford UniversityRafe MazzeoProfessor of MathematicsStanford UniversityJohn McCarthyProfessor of Computer ScienceStanford UniversityJohn D. McCarthyProfessor of MathematicsMichigan State UniversityJohn E. McCarthyProfessor of MathematicsWashington UniversityHenry P. McKeanProfessor of MathematicsCourant InstituteNew York UniversityMichael McKeownProfessor of Medical ScienceProgram in Molecular Biology, Cell Biology and BiochemistryBrown UniversityFormer Member - San Diego Unified Math Standards CommitteeFormer Member - Superintendent's Math Advisory Committee, San DiegoCo-Founder Mathematically CorrectMarc MehlmanAssociate Professor of MathematicsUniversity of Pittsburgh, JohnstownAdrian L. MelottProfessor of Physics and AstronomyUniversity of KansasAida MetzenbergAssistant Professor of BiologyCalifornia State University, NorthridgeStan MetzenbergAssistant Professor of BiologyCalifornia State University, NorthridgeM. Eugene MeyerProfessor of MathematicsCalifornia State University, ChicoJames E. MidgleyProfessor of Physics, EmeritusUniversity of Texas at DallasDragan MilicicProfessor of MathematicsUniversity of UtahHenri MoscoviciProfessor of MathematicsThe Ohio State UniversityClay Mathematics Institute ScholarGovind S. MudholkarProfessor of Statistics and BiostatisticsUniversity of RochesterGregory NaberProfessor of MathematicsCalifornia State University, ChicoBruno NachtergaeleAssociate Professor of MathematicsUniversity of California, DavisChiara R. NappiVisiting Professor of PhysicsUniversity of Southern CaliforniaOn leave from theInstitute for Advanced Study at PrincetonAnil NerodeGoldwin Smith Professor of MathematicsCornell UniversityCharles M. NewmanProfessor and Chair of MathematicsCourant Institute of Mathematical SciencesNew York UniversityLouis NirenbergProfessor of MathematicsCourant Institute, New York UniversityMaria Helena NoronhaProfessor of MathematicsCalifornia State University, NorthridgeRobert H. O'Bannon, Ph.D.Professor, Department of Natural Sciences and MathematicsLee UniversityCleveland, TNRichard PalaisProfessor of Mathematics, EmeritusBrandeis UniversityDimitri A. PapanastassiouFaculty Associate in GeochemistryCaltechThomas H. ParkerProfessor of MathematicsMichigan State UniversityDonald S. PassmanProfessor of MathematicsUniversity of Wisconsin at MadisonPeter PetersenUndergraduate Vice Chair and Professor of MathematicsDepartment of MathematicsUCLASteven PinkerProfessor of PsychologyDepartment of Brain and Cognitive SciencesMassachusetts Institute of TechnologyAuthor of How the Mind WorksJacek PolewczakProfessor of MathematicsCalifornia State University, NorthridgeDr. Ned PriceMathematics DepartmentFramingham State CollegeFramingham,Ma.David ProtasProfessor of MathematicsCalifornia State University, NorthridgeRalph A. RaimiProfessor Emeritus of MathematicsUniversity of Rochester, Rochester, New YorkDouglas C. RavenelProfessor and Chair of MathematicsUniversity of RochesterMarc A. RieffelProfessor of MathematicsUniversity of California, BerkeleyTom RobyAssistant Professor of MathematicsCalifornia State University, HaywardCris T. RoosenraadProfessor of MathematicsCarleton CollegeJerry RosenProfessor of MathematicsCalifornia State University, NorthridgeMary RosenProfessor of MathematicsCalifornia State University, NorthridgeYoram SagherProf. of MathematicsUniversity of Illinois at ChicagoCharles G. SammisProfessor of GeophysicsUniversity of Southern CaliforniaMark SapirProfessor of MathematicsVanderbilt UniversityPeter SarnakProfessor of MathematicsPrinceton UniversityStephen Scheinberg, Ph.D., M.D.Professor of MathematicsClinical Assistant Professor of DermatologyUniversity of California, IrvineWilfried SchmidDwight Parker Robinson Professor of MathematicsHarvard UniversityDr. Martha SchwartzGeophysicistCalifornia Mathematics Framework CommitteeCo-founder of Mathematically CorrectAlbert SchwarzProfessor of MathematicsUniversity of California, DavisRoger ShouseAsst. Professor of Education Policy StudiesThe Pennsylvania State UniversityBarry SimonI.B.M. Professor of Mathematics and Theoretical PhysicsChair, Department of MathematicsCaltechLeon SimonProfessor of Mathematics and ChairmanDepartment of MathematicsStanford UniversityDavid SingerProfessor of MathematicsCase Western Reserve UniversityWilliam T. SleddProfessor of MathematicsMichigan State UniversityAlan SokalProfessor of PhysicsNew York UniversityM.C. StanleyProfessor of MathematicsSan Jose State UniversityDennis StantonProfessor of MathematicsUniversity of MinnesotaProfessor James D. Stein Jr.Department of MathematicsCalifornia State University, Long BeachSherman SteinProfessor Emeritus of MathematicsUniversity of California at DavisHarold StevensonProfessor of PsychologyUniversity of Michigan, Ann ArborJ. E. StoneProfessor of Human Development & LearningCollege of EducationEast Tennessee State UniversitySandra StotskyDeputy Commissioner for Academic Affairs and PlanningMassachusetts Department of EducationResearch AssociateHarvard Graduate School of EducationRobert S. StrichartzProfessor of MathematicsCornell UniversityDaniel W. StroockProfessor of MathematicsMITJustine SuProfessor of EducationDirector, The China InstituteCalifornia State University, NorthridgeP. K. SubramanianProfessor of Mathematics & Computer SciencesCalifornia State University, Los AngelesHoward SwannProfessor of Mathematics and Computer ScienceSan Jose State UniversityDaniel B. SzyldProfessor of MathematicsTemple University, PhiladelphiaProfessor Sara G. Tarver, Ph.D.Department of Rehabilitation Psychology and Special EducationUniversity of Wisconsin-MadisonClifford H. TaubesDepartment of MathematicsHarvard UniversityAbigail ThompsonProfessor of MathematicsUniversity of California, DavisJohn B. WagonerProfessor of MathematicsUniversity of California at BerkeleyBertram WalshProfessor of MathematicsRutgers University--New BrunswickSteven WeinbergJosey Regental Professor of ScienceUniversity of Texas at Austin1979 Nobel Prize in PhysicsSteven H. WeintraubProfessor of MathematicsLouisiana State UniversityJames E. WestProfessor of MathematicsCornell UniversityBrian WhiteProfessor of MathematicsStanford UniversityProfessor Olof B. WidlundCourant Institute of Mathematical SciencesNew York UniversityHerbert S. WilfThomas A. Scott Professor of MathematicsUniversity of PennsylvaniaRobert F. WilliamsProfessor of Mathematics, EmeritusUniversity of Texas at AustinW. Stephen WilsonProfessor of MathematicsJohns Hopkins UniversityJet WimpProfessor of MathematicsDrexel UniversityCharles N. Winton, ProfessorDepartment of Computer and Information SciencesUniversity of North FloridaEdward WittenProfessor of PhysicsInstitute for Advanced Study at PrincetonJon WolfsonProfessor of MathematicsMichigan State UniversityWei-Shih YangProfessor of MathematicsTemple UniversityShing-Tung YauProfessor of MathematicsHarvard UniversityCalifornia dropped these math textbooks as have quite a few other states. Over 200 mathematicians across the nation in 2003 wrote then-Sec. of Education Richard Riley to complain about them. The Wall St. Journal bashed them in June 2000.

These books, and another series called "Investigations" for k-6 have been labelled "whole math," "fuzzy math" and some in AL are calling this "transformational math." When some gentlemen recently saw two of the Connected Math workbooks entitled, "What Do you Expect?: Probability and Expected Value" and "How Likely is It?" (both used in middle school grades), they asked whether the Ala. schools were getting the students prepared for careers in gambling! One workbook's cover had a roulette wheel and a Queen of Hearts card; the other included a quarter and three dice. Maybe these men are on to something because a few years ago, Mississippi authorized courses in gambling at one its state run colleges to prepare students for casino jobs.

An Evaluation of CMPR. James Milgram This report considers the National Science Foundation sponsored middle school mathematics program, CMP, published by Dale Seymour Publishers, and developed by G. Lappan and others, primarily at Michigan State University.If one visits the web site of the program, http://www.math.msu.edu/cmp/Index.html, one finds two preprints, presumably using rigorous methodology and statistical analysis, that are advertised as showing the benefits of CMP. Unfortunately, as we see in the appendix to this report, both studies are fatally flawed and deceptively presented. Additionally, at the website one will find a strong endorsement of the program by the AAAC. They grade it as one of the most effective programs for teaching middle school matematics Unfortunatly, this too must be taken with a grain of salt, as is also discussed in the appendix. In fact, it is generally acknowledged that there are no reputable studies showing that any of the NSF developed mathematics programs actually benefit students in testable ways.Leaving aside these issues, we turn to the program itself.Connected Mathematics Project consists of eight reasonably short booklets for each of grades six, seven, and eight. The booklets for grade six are as follows:1) Prime Time -- factors and multiples2) Data About Us -- statistics3) Shapes and Designs -- two-dimensional geometry4) Bits and Pieces I -- understanding rational numbers5) Covering and Surrounding -- two-dimensional measurement6) How Likely is It? -- probability7) Bits and Pieces II -- using rational numbers8) Ruins of Montarek -- spatial visualizationThe booklets for grade seven are:1) Variables and Patterns -- introducing algebra2) Stretching and Shrinking -- similarity3) Comparing and Scaling -- ration, proportion, and percent4) Accentuate the Negative -- integers5) Moving Straight Ahead -- linear relationships6) Filling and Wrapping -- three-dimensional measurement7) What Do You Expect? -- probability and expected value8) Data Around Us -- number senseThe booklets for grade eight are:1) Thinking with Mathematical Models -- representing relationships2) Looking for Pythagoras -- the Pythagorean theorem3) Growing, Growing, Growing -- exponential relationships4) Frogs, Fleas, and Painted Cubes -- quadratics relationships5) Say It with Symbols -- algebraic reasoning6) Kaleidoscopes, Hubcaps, and Mirrors -- symmetry and transformations7) Samples and Populations -- data and statistics8) Clever Counting -- combinatoricsOverall conclusionsOverall, the program seems to be very incomplete, and I would judge that it is aimed at underachieving students rather than normal or higher achieving students. In itself this is not a problem unless, as is the case, the program is advertised as being designed for all students. In fact, as indicated, there is no reputable research at all which supports this.The philosophy used throughout the program is that the students should entirely construct their own knowledge and that calculators are to always be available for calculation. This means thatstandard algorithms are never introduced, not even for adding, subtracting, multiplying and dividing fractionsprecise definitions are never givenrepetitive practice for developing skills, such as basic manipulative skills is never given. Consequently, in the seventh and eighth grade booklets on algebra, there is no development of the standard skills needed to solve linear equations, no practice with simplifying polynomials or quotients of polynomials, no discussion of things as basic as the standard exponent rulesthroughout the booklets, topics are introduced, usually in a single problem and almost always indirectly -- topics which, in traditional texts are basic and will have an entire chapter devoted to them -- and then are dropped, never to be mentioned again. (Examples will be given throughout the detailed analysis which follows.)in the booklets on probability and data analysis a huge amount of time is spent learning rather esoteric methods for representing data, such as stem and leaf plots, and very little attention is paid to topics like the use and misuse of statistics. Statistics, in and of itself, is not that important in terms of mathematical development. The main reason it is in the curriculum is to provide students with the means to understand common uses of statistics and to be able to understand when statistical arguments are being used correctly. The first four bulleted items above, particularly the second and third, indicate areas where the program does not do an adequate job of developing basic skills necessary for students to continue with more advanced work in mathematics, leading to possible careers in technical areas. But even the first cannot be ignored. It is true that the standard algorithms are not the only methods for teaching standard computational skills, but, the skills associated with these algorithms -- see the reviewer's discussion of long-division, for example http://www.csun.edu/~hcbio027/standards/conference.html/may21/milgram.html -- as well as some training in proving algorithms correct must be developed within the program if one is to accept the idea that students will strictly construct their own methods. CMP simply does not do this.Also, as noted -- while most of the topics to which the fourth bullet is applicable are not essential for people who will never use mathematics seriously in their professions -- for students intending careers where mathematics is heavily used these topics can be essential.In the detailed analysis which follows we will study three aspects of the program. First we will look at most of the booklets for the sixth grade. Then we will follow one important subject, exponents and exponentials, which is primarily concentrated in the eighth grade material, and finally we will make some brief remarks about how the program handles graphing.The sixth grade texts:We begin our analysis of the program with the sixth grade texts. As we go through these booklets and a few of the more advanced ones, I will constantly be pointing out areas where the problems above occur. This is to help make the point that these are not isolated instances, but represent a consistent point of view towards the material, and the level at which it should be addressed. In fact, except for a very few instances, I do not try to locate and point out outright errors -- though there are a number -- since errors are inevitable in the first versions of any program, and what concerns us here are the teaching methods and objectives.The first of the sixth grade booklets is Prime Time. This booklet is concerned with prime factorization of whole numbers. In standards based curricula, such as that in California, this is a fourth and fifth grade topic (California Mathematics Standards, Grade 4, Number Sense, 4.1, 4.2, and Grade 5, Number Sense, 1.3, 1.4), but since I view the program as largely remedial, this is not to be regarded as a criticism.Prime Time begins by assigning a unit project to be handed in or reported on at the end of the unit. This project is worth noting -- here it is.My Special NumberMany people have a number they find interesting. Choose a whole number between 10 and 100 that you especially like.In your journal* record your number* explain why you chose that number* list three or four mathematical things about your number* list three or four connections you can make between your number and your world.As you work through the investigations in Prime Time, you will learn lots of things about numbers. Think about how these new ideas apply to your special number, and add any new information about your number to your journal. You may want to designate one or two "special number" pages in your journal, where you can record this information. At the end of the unit, your teacher will ask you to find an interesting way to report to the class about your special number.From both a mathematical and pedagogical point of view this is unfortunate. Mathematically, no integers except perhaps 0, 1, and -1 are more significant than any others. And pedagogically, this reflects a poor point of view towards the development of the number system. If one prefers one whole number over another, think what a big door this opens for hating complicated fractions and even worse, irrational numbers. Basically, such a project appears to me to be totally unjustified except in remedial situations.In fact, this is doubly unfortunate, since, -- with exceptions that will be noted below, but which are more or less typical of books at this level today -- the overall discussion of numbers and their factorizations in this booklet is first rate. The authors have access to people who know a great deal about the subject and it shows here.The first section in Prime Time is entitled "The Factor Game." This and the second section "The Product Game", are about as good an introduction to factoring whole numbers as I've seen. As their names imply these are games that the students play with each other where winning or losing depends on the structure of the factors in the numbers one starts with. However, already in the second section we see a problem. Venn diagrams are introduced towards the end of section two. But they are explicitly limited only to the set of factors of two numbers, with the intersection region labeled by the factors common to both. It seems to us that this is simply too limiting. Their introduction in this way and at this point is fine. However, it is hard to understand why there is absolutely no indication or exercise showing that they occur in contexts other than common divisors.This tendency of the authors to introduce important concepts and then leave them only as tantalizing fragments will become more and more common throughout the remaining booklets. This might be acceptable if, at least, in the teachers manual further details were given or indications of where to find more information, but this does not seem to be the case.The third section, "Factor Pairs", which uses area as an interpretation of factoring a whole number into two parts, is not quite as strong as the first two. For example, after looking at the rectangles such as 3 by 4 and 2 by 6 obtained by factoring 12, it might be natural to draw the conclusion that any time that a whole number is factored into a product of two whole numbers one can draw a rectangle with the whole number as area. It would even be natural to observe that the perimeters of such rectangles will generally not be the same. After all, these are both fourth grade standards in California (Grade Four, Measurement and Geometry 1.1 and 1.2). But no inferences whatsoever are explicitly drawn, either in the teachers manual or the student manual.At the end of section 4, on page 43, particularly problems 19 and 20, an excellent explanation of the sums n2 = 1 + 3 + 5 + ... + (2n+1), and 2(1 + 2 + ... + n) = n(n + 1) is given. But once more, the general result is never stated. In the teacher's manual, however, it is sort of stated, but not in a way that will help an inexperienced teacher to assure that the students do not miss the point. To make this clear, here are the comments in the teachers manual for these problems:19b. 1 + 3 +5+. .. + 39 = 400.19c. row 24: 47; The sum will be 576 in row 24,because 576 = 242. The last number in this row is 47 because 47 is the twenty-fourth odd number. This famous pattern is the sum of the consecutive odd numbers: the sum in each row is the square of the number of numbers in the row.20a. Tiles can be used to set up a visual display of this problem (see below left). From the pattern, you can see that adding the first n consecutive even numbers is the same as multiplying n times(n+1). So, the next four rows are as follows:2+4+6+8+10=30(which is 5 x 6)2+4+6+8+10+12 = 42(which is 6 x 7)2+4+6+8+10+12+14 = 56 (which is 7 x 8)2+4+6+8+10+12+14+16 = 72 (which is 8 x 9)20b. 2+4+6+...+40 = 420 (which is 20 x 21)20c. row 10; 20, because 20 is the tenth even numberThe fifth section on factorization leads to "discovering" the fundamental theorem of arithmetic -- the unique decomposition of whole numbers into products of primes. But, as usual, the theorem is never stated in the student edition. This is particularly relevant because, though it is possible for the students to understand what this theorem means, at this stage it is impossible for them to have, in any way, shape or form, proved it.If students get the idea, based on the explorations they've made of the meaning of the fundamental theorem of arithmetic, that they can then use it without having been TOLD that it is, in fact, true in all cases, (and that if they stick with mathematics long enough, they'll be given -- or construct -- a proof), then they will have learned something VERY VERY DANGEROUS.All too often we see students at the most advanced levels use results that are only partially true as though they were true in every case, and serious problems can and do result from this. But, as indicated, this is the approach taken throughout the three year CMP sequence. I was never able to find a place where students were warned that something which appeared to be true after a large number of trials might fail after even more trials. Likewise, I was unable to find any point in the program where any statement that had been verified by the students for a number of cases was proved true in all cases. We will discuss this further when we discuss the programs treatment of algorithms in the seventh booklet,Addendum: Recently a colleague who's son is currently in sixth grade in a school system that uses CMP exclusively pointed out a very serious difficulty with this booklet that I had not noticed originally. The material here is not well understood by many sixth grade teachers, and the discovery method that is used, never stating what the objective of each lesson is, applies to the teachers manual as well. The material is not explained there and the objectives are not stated there. I di not take this into account when reading the teachers manuals since both of these are clear to someone who knows the material very well.That was not the case in this class in the Palo Alto school system: the teacher seems to have had only the fuzziest idea of what the real objectives of the material were, so the students dutifully did the exercises without any guidance and developed no insight into what was happening. The result was that the students were totally unable, by the end of the book to make any sense at all of the locker problem.It is critical, and even understood in a kind of general way by most mathematics educators, that teachers must understand the material even better in a discovery situation than they need to when the instruction and material are more traditional.Bits and PiecesIt would also have been natural at this point to introduce exponents -- which is a fifth grade standard in California (Grade 5, Number Sense, 1.3) -- but this is not done. In fact the first mention that I was able to find of exponential notation occurs on page 42 of the final seventh grade booklet, Data Around Us.Finally, the sixth section, the locker problem is excellent. However, here, relating the evenness or oddness of the number of factors of a number to the result in the end absolutely begs for further examples, such as, e.g., the Konigsberg bridge problem. The point is that the process of understanding why only certain lockers are left open after a number of students have passed through is a pure process of abstraction. The method of thought implicit here is the same method used to solve the Konigsberg bridge problem, though the contexts initially appear to be totally different. This is another of these examples where something good is started but left hanging.Overall, though, this is a good set of lessons. In a curriculum such as that in California it could be used to good effect as a supplement for the normal fifth grade material, and to help with remediation in the sixth grade. In these contexts the failings noted above would not be significant. But Prime Time is also the high point of the five sixth grade booklets that I have examined. The others range from significantly less good to very bad indeed, as the problems noted above become more and more significant.The next booklet that we shall examine is"Data About Us".The first section works well as an introduction to the subject of data collection.The second section, "Types of Data""starts out badly however. The second paragraph reads:When we collect data, we are collecting a measurement" about some "thing." We are interested in organizing the data by tallying, or finding the frequency of occurrence for each data value." Of course, data is data. We introduce measurement as a means of describing data.Except for this, the material is sound, and, overall the material is well covered. However, one should be aware that this material, in California at least, and certainly in countries like Japan and Singapore is covered one to two years earlier than sixth grade. (California Math Standards, Grade 4, Statistics, Data Analysis, and Probability, 1.1, 1.2, 1.3 and Grade 5, 1.1, 1.2, and 1.3) In fact the standards mentioned go significantly further than the material covered in Data About Us. A characteristic of the discussion throughout, is a superficial analysis of data and simply a description of the meanings of basic terms, but no precise definitions. In the student material for the fifth section, "What do we mean by the mean, the term mean is NEVER defined explicitly.In summary, this booklet has a distinctly remedial character and, while the material is well organized and important, it is treated purely descriptively, not precisely. Terms, even basic terms like mean, are never defined.Here, even more so than with Prime Time, one can imagine that the best use of this material is in a situation where the students are distinctly behind their expected level and have not had any experience of the precision of mathematics. In fact, in the local district where I live, both Data About Us and Prime Time were used in exactly this kind of situation -- a sixth grade class where the students had used Mathland in their earlier grades and had, consequently, extremely weak basic skills. In this situation the CMP booklets were very well received by the students, and the teachers reported that the students skill levels improved dramatically.Addendum: In the case of the Palo Alto sixth grade class, the results with this text were also very disappointing. The lacks mentioned above were magnified by the lack of understanding of the material by the teacher and the failure of the teachers manual to offer any detailed help. Once more it is reported that the result was complete confusion on the part of the the students. By contrast, in the class in my local district, it was evident that the teacher knew the material throughly, commenting repeatedly when discussing both this booklet and the previous one with me, that she had known the material in a general way previously, but that seeing it developed in this way, and having to reconstruct it for herself, clarified it for her enormously. We can infer from this that it was the process of understanding going on with this teacher which enabled her to be successful in using the booklet with her class.It appears essential to reiterate my previous observation that in a discovery situation it is absolutely necessary that the teacher understand the material considerably better than is required in a more traditional environment. It is likewise necessary to note that it is totally unrealistic to expect this from the vast majority of the teachers in this nations middle schools.The next book in the series is Shapes and Designs. Here the remedial nature of the program becomes even clearer. What follows is the description of the individual sections from the teachers manual. Investigation 1: Bees and PolygonsThis investigation poses the key question, What tile shapes can be used to cover the plane? It asks students to make conjectures about why honeycombs are covered with hexagons and to use physical materials to explore other possibilities.Investigation 2: Building PolygonsThis investigation is based on the general question, Is the shape of a polygon determined exactly by the lengths of its sides and the order in which those sides are connected? The three problems involve the use of manipulatives called Polystrips.Investigation 3: Polygons and AnglesThis investigation introduces three basic ways of thinking about angles and the ideas behind angle measurement. It gives students practice in estimating angle measurements based on a right angle. A measuring device, the angle ruler is introduced, allowing more precise measures of angles. Students then explore a problem that looks at the possible consequences of making measurement errors.Investigation 4: Polygon Properties and TilingThis investigation focuses attention on some basic properties of familiar quadrilaterals, using tiling as a context.Investigation 5; Side-Angle-Shape ConnectionsIn this investigation, students look at what remains constant and what changes as triangles, squares, rectangles, and parallelograms are rotated and flipped. The symmetries of the figures become more evident as students work with them.Investigation 6; Turtle TracksIn this investigation, students use the logo programming language to create computer designs, two of the three problems in this investigation can be done even if students do not have access to computers.As an example of level, compare the California standards (Grade four, Measurement and Geometry, 3.3, 3.4, 3.5, and 3.6 as well as the fifth grade Measurement and Geometry standards, 2.1, 2.2, and 2.3). Indeed, in checking the further geometry booklets in grades seven and eight of the CMP program, we find that in total, they cover no more than the material spelled out in these fourth and fifth grade standards.The discussion in Shapes and Design ultimately focuses on angle measure but never really states anything specifically. A great deal of time is spent MEASURING angles with a protractor, to the point where several problems are given on page 38 listing series of measurements of the same angle and asking questions like ""What method would you use to decide on the best measurement for each angle?" This does not seem to be well designed as preparation for geometry where the focus is on the abstract properties of precisely known angles and lengths. Moreover, though it is assumed in the teachers' manual that the teacher knows that the sum of the interior angles of a triangle is 180 degrees, I searched in vain for any explicit statement of this in the student material. The students are, presumably, supposed to figure this out for themselves with their inaccurate angle measurements.It is precisely at this point that CMP becomes strictly remedial. If students are to go on to higher levels of achievement in mathematics, from geometry through calculus, linear algebra and beyond, they must be able to handle precisely defined abstract concepts. Moreover, these abilities are difficult for even the strongest students to master, and they take considerable time to develop.Pages 40 and 41 in Shapes and Designs are very bad with respect to the considerations above. Here definitions are confused with measurements in the case of the angles that a transverse line makes with "parallel" lines. (The point is that "parallel" is really an abstract concept, and we cannot decide if two lines are parallel by "real" measurements which always have some errors.)Incidentally, at this point, all the authors had to do was just mention as a fact that the angles of intersection of transversals with parallel lines are the same, and all the material needed to demonstrate that the sum of the interior angles of the triangle is 180 degrees would have been available. But as I've indicated is typical in this program, they promptly leave the subject hanging on page 40 and don't seem to return to it again in this booklet. However, on page 50 of Shapes and Designs we find the following:These questions will help you summarize what you have learned:a. In regular polygons, what patterns relate the number of sides to the angle sum and the size of the interior angles?b. In irregular polygons, what patterns relate the number of sides to the angle sam and the size of the interior angles?Think about your answers to these questions, discuss your ideas with other students and your teacher, and then write a summary of your findings in your journal.The final section, "Turtle Tracks", uses turtle graphics through logo to give students some experience with computer programming. By comparison, very similar problems occur in the third and fourth grade texts of the McGraw-Hill SRA series,"Explorations.Here is another problem I had with this booklet. On page 21c there is the following sidebar:For the Teacher: Generalizing Mathematical StatementsSome teachers take this opportunity to discuss with students howmathematicians think and how they record the results of theirexperimentation:"This is not something you are responsible for knowing, but I want toshow you how mathematicians would use the language of mathematicsto record your generalization. Mathematicians try to talk about ideas ata general level, rather than about a specific case. For example, they givenames to the lengths of a triangle's sides rather than talking about atriangle with specific sides like 8 cm, 5 cm, and 6 cm. Instead, they callthe sides of a triangle by letters, such as side a, side b, and side c. So atriangle with sides a, b, and c stands for any triangle you can make.Mathematicians would write your statement like this:"If a and b represent the two shorter sides of a triangle and c representsthe longest side, then a + b > c. In one sense, this is the beginning of something potentially very important -- an effective introduction of the process of abstraction. But note how it is introduced: students could be made aware that "mathematicians" think about things in this way. There is no indication that students could benefit by trying to think in this way. In another sense this is not by any means an accurate description of either mathematicians or the process of doing mathematics. Things are stated in generality only when the statement is sufficiently important and useful that a general statement is merited. There are innumerable papers in the mathematical literature giving detailed analyses of single examples.We now turn to the seventh booklet, Bits and Pieces II, which is the second booklet discussing rational numbers. Here are the overview and the the author's description of the mathematics in this booklet.Rational numbers are the heart of the middle-grades experiences with number concepts. Fromclassroom experience, we know that the concepts of fractions, decimals, and percents can bedifficult for students. From research on student learning, we know that part of the reason forstudents' confusion about rational numbers is a result of the rush to symbol manipulation withfractions and decimals.In Bits and Pieces I, the first unit on rational numbers, the investigations asked students tomake sense of the meaning of fractions, decimals, and percents In different contexts. In Bits andPieces II, students will use these new numbers to help make sense of many different situations.The Mathematics in Bits and Pieces IIThis unit does not teach specific algorithms for working with rational numbers. Instead, it helpsthe teacher create a classroom environment where students consider interesting problems inwhich ideas of fractions, decimals, and percents are embedded. Students bump into these impor-tant ideas as they struggle to make sense of problem situations. As they work individually, ingroups, and as a whole class on the problems, they will find ways of thinking about and operat-ing with rational numbers.The teacher's role is to help students make explicit their growing ideas about the world of ratio-nal numbers and, when students are ready, to inject ideas and strategies into the conversationalong with the ideas and strategies generated by the students. Simply giving students algorithmsfor moving symbols for rational numbers around on paper would be a mistake and the tempta-tion to do so is often great. All teachers want their students to succeed, and showing them howto do something such as how to cross multiply to compare two fractions gives the impres-sion of immediate success. Students can do the algorithm by memorizing. However, evidencefrom student assessments shows that students do not understand algorithms that are given tothem in this way and therefore cannot remember or figure out what to do in a given situation.This unit provides a rich set of experiences that focus on developing meaning for computationswith rational numbers. We expect students to finish this unit knowing algorithms for computa-tion that they understand and can use with facility.The discussion above of the mathematics in Bits and Pieces II represents a highly controversial point of view about the subject. This view is agreed with by less than 1% of the professional mathematicians in California, for example. No one would dispute the argument that rote memorization of algorithms alone does not lead to understanding. However, when an algorithm is introduced together with a careful and precise explanation of how and why it works, students are exposed to material that is critical to the continued development of their mathematical skills. Whether students learn these types of things using discovery methods or other methods is not important. What is critical is that they learn them somehow.Now we turn to the individual sections of Bits and Pieces II.The first section discusses percents, and concentrates on percent reductions in prices, sales taxes, and tips. It is grade appropriate and solid mathematically. It is, in fact, among the best presentations of this topic at the sixth grade level that I've seen in the sense that the ideas are clearly explained and evidently understood by the authors. However, the material here is not really theory. These are concepts that play a major role in everyday life. It is a start (and an important one) that the concepts be clearly enunciated. But these topics demand mastery level learning on the part of the students.At this point the discovery method and nothing else philosophy of the authors definitely works to the detriment of the students. The numerical skills involved with these topics require practice on the students' part, and discovery methods do not encourage this. Indeed, the discovery approach is carried to extreme levels here. For example note the quote on page 164 of the teachers manual in a sample letter meant to be sent to parents: "It is important that you do not show your child rules or formulas for working with fractions. This unit helps students to discover these rules for themselves . . . ." So, in spite of the basically solid exposition, if the subject is taught as the authors seem to intend, there is every reason to expect that the students will not learn the material to the depth required.The continuation in the second section maintains the high level of exposition found in the first. Here percents are also tied in to topics in the data analysis and statistics strand. The reservations indicated above are less compelling here as the material is not quite as basic.The third section is concerned with estimating using fractions and decimals. The discussion is developed through the notion of deciding when a benchmark fraction represented on the number line is closest to another number. But in the first example, they already show a difficulty with this by using an example where the number is exactly half-way between the two closest benchmark numbers. As usual, however, they leave this hanging. Then these benchmark numbers are used to estimate sums of fractions and a game, "Getting Close" is introduced. A large number of problems involved with various aspects of estimation are then given, both numeric and geometric.My personal view is that too much time is devoted to estimation. The costs here are in diluting the precision of the abstract concept of a fraction with the approximate nature of many "real world" applications. One of the chief aims of a traditional education in mathematics was to give students experience with precise thinking -- the ultimate aim being to aid them in making reasoned, rational decisions -- and the approach to manipulating fractions here is not well aligned with that objective. But this is a matter of opinion and should not exactly be taken as an objection.The third section is well done, taking its objectives into account. But it is important to realize that my view of these objectives is that they are aimed at weaker students. Diluting the precision of the concept of a fraction cannot possibly be of help to students intending to go on to study advanced topics in mathematics such as those required for careers in engineering, economics, or other related technical areas. Here students must be prepared to deal on an everyday basis with things like Laplace transforms which convert systems of linear differential equations into matrices whose entries are quotients of polynomials. If the concept of a fraction is not crystal clear to these students, they will have severe difficulties at this point. Indeed, too often in recent years, this is exactly what I've seen even in classes at Stanford.The fourth section is a different story. Here the authors mix approximation with exact numbers, totally confusing the two through the means of a land map, which describes, using straight lines but no indications of length or area, the decomposition of two sections of land among many owners. Then it is assumed that various sales took place and precise contiguous areas e.g., 1/2 of one section are supposed to have ended up in the hands of only four of the original owners. This mix of precision and imprecision is never clarified and is used as the basis for explaining addition and subtraction of fractions.At this point, consistent with the point of view towards algorithms described in the introduction to this booklet, the students are asked to work in groups and find their own algorithms for adding and subtracting fractions. Moreover, as verifications of correctness they are given the following instructions:Test your algorithms on a few problems.... If necessary, make adjustments to you algorithms until you think that will work all the time. Write up a final version of each algorithm. Make sure they are neat and precise so others can follow them." The simplest method for achieving this is to simply make a list of the test problems and their answers, and the algorithm would be -- locate the set problem on the list and write down the answer.But to add insult to the anti-mathematics above, on page 48, the same page as the directions above, the following definition of an algorithm is given:To become skillful at handling situations that call for the addition and subtraction of fractions, you need a good plan for carrying out your computations. In mathematics, a plan -- or a series of steps -- for doing a computation is called an algorithm. For an algorithm to be useful, each step should be clear and precise so that other people will be able to carry out the steps and get correct answers. This is incorrect. What they have defined is a "program," and even this is not quite right. A central issue in the development of both programs and algorithms is the issue of correctness. This is subtle but crucial. If students are going to develop programs and algorithms but are never shown how to prove them correct, disasters occur. On the other hand, at this level it may be very difficult to demonstrate the correctness -- or more likely, incorrectness -- of a student provided algorithm.Approximately 1 in 10 of the students I've had in recent years in the differential equations course at Stanford have believed that (a/b) + (c/d) = (a+c)/(b+d). It is somewhat difficult to be comfortable with an engineer who does his/her calculations in this manner.In the fifth section, to illustrate that the material above was no accident, it is proposed, on page 59 that the students work in groups to develop their own algorithms for multiplying fractions. The test of correctness is a repeat of the directions above.The final section, "Computing with Decimals" is better, assuming that the students have survived sections 4 and 5 and actually have correct methods for adding, subtracting, multiplying and, as they say in the introduction "possibly dividing decimals." The explanations and exercises here seem to actually be helpful.But as is becoming more and more the norm with this program, there is a difficulty. Here is the main part of page 69:When you multiply 0.1 by 0.1 on your calculator, you get 0.01. What is the fraction name for 0.01? It is 1/100, as you saw with the grid model.In the next problem, you explore what happens when you multiply decimals on your calculator. Before you use a calculator to find an exact answer, think about how big you expect the answer to be..Problem 6.3A. Look at each set of multiplication problems below. Estimate how large you expect the answer to each problem to be. Will the answer be larger or smaller than 1? Will it be larger or smaller that 1/2?Set 121x1 =21 x 0.1 = etc.21 x 0.01 =21 x 0.001 =21 x 0.0001 =B. Use your calculator to do the multiplication, and record the answers in an organized way so that you can look for patterns. Describe any patterns that you see.Now we see why the authors believe that they can proceed in the way indicated. The students do not, in fact, have to learn to actually add, subtract, multiply, or above all, divide decimals, since their calculators will do it for them.Unfortunately, as has been shown in work on curricular development, many of the cognate skills implicit in things like learning the long division algorithm become important in different contexts many years later. Consequently, students who have not developed these skills often seem to find themselves at a serious disadvantage when attempting to work in technical fields.All in all, Bits and Pieces II, in spite of the initially good exposition in the first two sections and part of the last, is a very poor booklet, and probably does more harm than good.The last booklet in the series for sixth grade is Ruins of Montarek. Here is the last paragraph of the overview in the teachers manual. It is clear from this paragraph that the material here is entirely remedial, and this is consistent with the content which correspond with the California Third Grade Standards (Measurement and Geometry, 1.1, 1.2)Spatial visualization skills are very important in developing mathematical thinking and are critical to reading graphical information, using arrays and networks, and understanding the fundamental ideas of calculus. In the past several decades, research has raised many questions about spatial visualization abilities. Many studies have found that girls do not reason as well about spatial experiences as do boys, especially starting at about adolescence. The explanation offered by some psychologists, that this difference may be innate, is unacceptable to those of us concerned with teaching children. It might also be worth noting that the role of spatial visualization in topics such as calculus is overstated. In fact the vast majority of the topic takes place in two dimensions. When one finally gets to questions in three dimensions involving solid integrals, surface area, and related topics, virtually all students appear to have serious difficulties, and the "skills" developed in this booklet are not going to address the problem areas which actually occur.This completes our review of the sixth grade material in this program. Throughout, our perspective has been to illustrate the ways in which it differs in essential ways from more standard programs. We now turn to a discussion of some of the eighth grade material to illustrate the fact that these differences remain consistent throughout the entire program. The handling of exponents and exponentials in CMPIn traditional texts, the Japanese texts, and most others, the order in which exponentials is done is to first introduce exponential notation, explain the exponent rules in the case where the exponents are positive integers, explain that a0 = 1, and using this, explain that a-n = 1/an. After this students learn about fractional exponents. (Of course it helps enormously if the students have already discussed topics like square roots and maybe cube roots.) Finally, if there is time, the general form ax is introduced for any number x (and positive a). The cognate topics here that it is natural and perhaps necessary to discuss are things like the existence of irrational numbers and the fact that numbers like the square root of 2 are irrational.Also, traditionally, one of the main methods of introducing exponents was through compound interest, a standard seventh grade topic.In short, developing a reasonably full discussion of exponential relationships involves a major effort.The booklet Growing, Growing, Growing is the only one among the 24 booklets to discuss any of these topics with the exception of Frogs, Fleas, and Painted Cubes, which discusses quadratics but is recommended for discussion after Growing, Growing, Growing, and, in Thinking with Mathematical Models which precedes Growing, Growing, Growing, a discussion of functions of the form a + bx-1, (though written in the form a + b/x), and a single example of compound interest (again done without using the exponential notation).Consequently, before we discuss Growing, Growing, Growing in detail we will briefly discuss the two sections in Thinking with Mathematical Models which involve exponentials. In section two, "non-linear models," the discussion starts with a physics experiment using beams made of paper. The students are to suspend the beams at their endpoints and successively pile pennies on the center till the beams crumple. They then graph the results and, hopefully, notice that the result is non-linear. No discussion of the physics involved is given, nor could there be at this level. Finally, the discussion focuses on the equations above, and the general shape of the graphs are described.The third section, "More Nonlinear Models" in Thinking with Mathematical Models, starts with an elementary example of compound interest. Then it turns to another experiment -- this time with a glass filled with water. Half the water in the first glass is poured into a second glass, half the water in the second glass is poured into a third, and so on. The students are supposed to then notice the inverse exponential shape of the resulting water levels. The remainder of the discussion here takes place in the problems.Both of these sections are entirely descriptive. The final problem in the third section will give an idea of the depth of the discussion.Four biologists are studying the caribou and wolf populations in a particular area of Alaska. The caribou are prey to the wolves, and keeping the two populations in balance is important to the survival of both species. The biologists are trying to predict what will happen if no measures are taken to control the populations. After studying the situation, each biologist makes a graph of his or her prediction. Describe what each graph represents in terms of the animal populations. The four graphs are respectively a straight line parallel to the x-axis, a mildly concave (downward) curve, a sharply concave curve but with the base labeled wolves and a straight line with negative slope. In all four cases the y-axes is labeled caribou, and in the remaining three cases the x-axis is labeled time.The actual mathematical issues involved -- for example the standard non-linear differential equations modeling predator-prey situations -- are far beyond the level of the students. Also, the answers given are not realistic. In point of fact, what tends to happen is that as the caribou population declines, the wolf population initially rises but then also declines which allows the caribou population to increase, and the situation cycles. (This is the situation near the stable point of the Volterra predator-prey equations. ) The situation of the first graph occurs only at the single fixed point, so should NEVER be observed in nature.Now let us turn to the booklet Growing, Growing, Growing. In a standard algebra course, one of the key topics is exponents. In fact, usually, exponents have been introduced in seventh grade, and maybe even sixth grade with squares and cubes being written in the exponent notation. Also, in seventh grade it is not unusual that some fractional exponents have been introduced -- at the least, square roots. In any case, in a standard course students are expected to learn and understand the exponent lawsa(m + n) = a m a n and amn = (a m) n. Here is the totality of the discussion of the exponent laws in Growing, Growing, Growing.2. Cesar said that since he can group 2x2x2x2x2x2x2x2x2x2 as(2x2x2x2) x (2x2x2x2x2x2), it must be true that 210 = 24 x 26a. Verify that Cesar is correct by evaluating both sides of the equation210 = 24 x 26.b. Use Cesar's idea of grouping factors to write three other expressions that are equivalent to 210. Evaluate each expression you find to verify that it is equivalent to 210c. The standard form for 27 is 128, and the standard form for 25 is 32. Usethese facts to evaluate 212. Show your work.d. Test Cesar's idea to see if it works for exponential expressions with other bases, such as 38 or 1.511 lfest several cases. Give an argument supporting your conclusion.e. Find a general way to express Cesar's idea in words and with symbols.Extensions_______13. Molly figured out that 26 = 64 and 43 = 64. Then, since 22 = 4, she substituted 22 for 4 in the expression 43 and got (22)3 = 64. She said that since 26 = 64 and(22)3 = 64, it must be true that (22)3 = 26.a. Verify that Molly is correct by evaluating both sides of the equation(22)3 = 26.b. Use Molly's idea to find an exponential expression equivalent to the given expression.(34)2(43)2c. Find a general way to express Molly's idea in words and with symbols. Check your idea by testing it on three more examples. In this instance, as is typical of the program, an extremely important topic is introduced and immediately dropped. Here is the next problem.14. Juan wrote out the first 12 powers of 2. He wrote 21 = 2,22 = 4, 23 = 8, and so on. He noticed a pattern in the digits in the units places of the results. He said he could use this pattern to predict the digit in the units place of 2100.a. What pattern did Juan observe?b. What digit is in the units place of 2100. Explain how you found your answer.Of course, what is going on here is arithmetic modulo 10. This is a much more sophisticated topic, and one seldom felt necessary to discuss in K - 12, since it has very limited applicability. Moreover, as is evident, no attempt is made to explain what is going on, and certainly the students are not asked to justify or prove their answers. They are expected, - as the "solution" in the teacher's guide shows - to notice after eight trials that there appears to be a repeating pattern 2, 4, 8, 6, 2, 4, 8, 6, and to guess that this is, in fact, the general case. Mathematically speaking this is a disaster, and would be unacceptable in a program designed for students requiring serious mathematical backgrounds.The next section, "Growth Patterns" is again purely descriptive. The basic intent is to introduce the concept that exponential growth is characterized by the property that the value at stage n is a constant times the value at stage (n-1). The material here serves a useful purpose as an introduction to exponential growth and decay. However, the actual mathematics involved in any kind of serious study of these topics is quite a bit more advanced, almost inevitably requiring calculus. Consequently, it is traditionally deferred to a much later stage in the curriculum. For example it is not discussed before grade 10 in the Japanese books discussed above. Indeed, the following problem in this second section illustrates the level of the discussion here:Calculators use scientific notation to display very large results. For each expression, find the largest whole-number value of n for which your calculator will display the result in standard notation.a. 3nb. pnc. 12nd. 237nThe next section "Growth Factors" considers exponential growth as before , the only difference being that now the multipliers are no longer required to be whole numbers. There is a good introduction to compound interest here, though only the growth of value of an investment is considered.Here is one of the final exercises in this section:If your calculator did not have an exponent key, you could evaluate 1.512 by entering 1.5 x 1.5 x 1.5 x 1.5 x 1.5 x 1.5 X 1.5 x 1.5 x 1.5 x 1.5 x 1.5 x 1.5.How could you evaluate 1.512 with fewer keystrokesWhat is the least number of times you could press x to evaluate 1.512? What is unexpected here is the answer to the second part given in the teachers manual:Answers will vary. The calculation given in the answer to the first part requires four presses of x, as does this calculation: 1.5 x 1.5 = 2.25, 2.25 x 2.25 x 2.25 = 11.390625, 11.390625 x 11.390625 = 129.7463379. This is nonsense. There can only be one correct answer to any question which asks for a least number. In fact, the minimum possible is four. On the other hand, this is another example of a puzzle problem in the book, which, as far as I can see, leads to no interesting or important developments in the subject. This concludes our discussion of the handling of exponents in CMP. Again, let me emphasize that for the most part what I've tried to do is to point out the distinction between the handling of this very important topic in CMP, and what would be expected in a more standard program, geared to developing skills needed in more advanced areas of mathematics.The handling of graphs in CMPLet us conclude this review by considering a crucial part of the treatment of graphs in the eighth grade component of CMP, and comparing it to the handling of this topic in the seventh grade Japanese texts.In the seventh grade algebra CMP text Moving Straight Ahead, graphs of linear equations are considered, and in a few places the intersections of the graphs of two different linear equations are considered, but no general methods seem to be introduced for determining the intersection. For example there is a related rate problem involving a race between two brothers, Henri and Emile, in 2.5. In 3.1, the booklet takes up the discussion of this probem again with the following remarks found on page 52h.The point of intersection is the point at which Emile overtakes Henri. The boys will be at the same distance from the starting line and will have walked the same amount of time.What are the coordinates of the point of intersection? (The intersection occurs at t=30 seconds and d = 75 meters.)How did you find the point of intersection?Students will have found this in Problem 2.5 by making a graph by hand.Explain that this point can also be found by using a graphing calculator. Enter the equations into your overhead graphing calculator (if you have one), and have students do the same. ...Incidentally, there is a curious problem on page 75. Problem 29 is given as follows:In 1980, the town of Rio Rancho, located on a mesa outside Santa Fe, New Mexico, was destined for obscurity. But as a result of hard work by its city officials, it began adding manufacturing jobs at a fast rate. As a result, the city's population grew 239% form 1980 to 1990, making Rio Rancho the fastest-growing "small city" in the United States. The population of Rio Rancho in 1990 was 37,000.a. What was the poplulation of Rio Rancho in 1980?b. If the same rate of population increase continues, what will the population be in the year 2000? The answer given for 29(a) is 2.39P = 37,000 so P = 15,481 people in 1980. I guess we should be glad, that the population did not increase 0%. To make it clear that this is not an accident, here is the answer for 29(b). P = 2.39(37,000) = 88,430 people in 2000. Actually, I had not initially noticed problem 29 because of this. The thing that initially interested me was that they had misplaced Rio Rancho by about 50 miles.The final section of A World of Patterns, is introduced as follows: "In this investigation you will sketch graphs that fit written descriptions, and you will make up stories about what a given graph might represent." Remember that this is supposed to be an eighth grade algebra text. This appears to be a discussion of graphs that would be more appropriate in a much lower grade, for example compare the California standard, (Grade five, Algebra and Functions, 1.1).To illustrate the low level of skills developed in this booklet consider problem 7 on page 55. In your previous math work, you investigated the relationships among the radius, height, base area, and volume of a cylinder. You found that the volume of a cylinder is equal to its base area multiplied by its height.a. Suppose you are in charge of designing a cylindrical can to hold 250 ml of juice. Investigate some possible (base area, height) combinations for the can. Try radii of 2.5 cm, 3 cm, 3.5 cm, and any other measurements you think are reasonable. Record your finding in a table. b. Make a graph of your (base area, height) data.c. Draw a straight line or curve to model the data. What other situations in this unit have similar graph models?d. Write an equation that fits your graph model.e. Which (base area, height) combination would you choose for the can? Give reason for your answer. Notice that there is no requirement at all that the students do any symbolic manipulations. In a more standard text here is what might be done. The volume is given by the formula V = (pi)r2h, and this is assumed constant. The students might well be asked to determine the surface area of a can with constant volume, which would be given by the formula 2(pi)(r2 + rh), so substituting for h,A = 2(pi)(r2 + V/((pi)r)).They could then graph this to estimate a the size of a can with minimal area. (It would be asking too much to have them determine the minimum exactly.) But this variant of the CMP problem is at a reasonable level for an eighth grade algebra text. It is also worth noting that the basic formulae for the area and volume suggested above are contained in the California sixth grade standards, (Measurement and Geometry, 1.2, 1.3, and Algebra and Functions, 3.1, 3.2).Let us consider, by comparison, the handling of the topic of graphs in the Japanese seventh grade text from 1984 translated and published by The University of Chicago School Mathematics Project in 1992.Their chapter on functions starts on page 97 in a similarly descriptive way with the description of the height of a meteorological rocket as a function of time. It then presents a precise definition of a function on page 99.When quantities vary in accordance with changes in other quantities, all these quantities are expressed as variables such as x and y. If we determine the value of x, the value of y is also determined. In situations like this, we say that y is a function of x. On page 101 it introduces proportions and inverse proportions with the following remark.You learned about proportions and inverse proportions in elementary school. Now we will learn about functions which are defined by proportions and inverse proportions. In the problems on functions and proportions on page 107 here is the second problem:The bottom of a rectangular container is 40cm long and 20cm wide. If we let 200 cm3 of water into the container every second for t seconds, the depth of the water becomes h cm. Answer the following problems:Express h in terms of t and show that h is proportional to t.Is t proportional to h? If so, state the constant of proportionality. Then a discussion of coordinates and graphs is initiated on page 108. Once more it is pointed out that graphs of linear equations had already been learned in elementary school, but only in the first quadrant, and that in this section they will extend their knowledge to negative numbers as well. The section concludes with a study of functions of the form y = ax-1, and the precise definition is given:Generally, when a is a nonzero constant, the graph of y = ax-1 consists of two smooth curves. This curve is called a hyperbola. The Japanese program is similar to CMP in these grades in that it is "integrated," and programs like CMP are supposedly developed on the Japanese model. However, as indicated above, there is a dramatic difference in level between the two programs. In terms of content and precision of the treatment of the various topics a traditional US program would tend to be much nearer to the Japanese model . The main difference is in the fact that the traditional US programs tend to cover many fewer topics each year in the higher grades.Graphs are also considered in a few of the other books, for example the eighth grade booklet Thinking with Mathematical Models and the seventh grade booklet, Variables and Patterns, but in all cases the discussion is at or below the level indicated above.Appendix: The supporting literature for CMPI started my evaluation of the middle school program -- Connected Mathematics Project -- by visiting their web-site http://www.educ.msu.edu/cmp/, and printing out the article Effects of the Connected Mathematics Project on Student Attainment by M.N. Hoover, J.S. Zawojewski , and J. Ridgway. There is a brief analysis of the article by W. Bishop that can be found at http://lynch.nscl.msu.edu/tsang/eval1.htm.Perhaps the most interesting and important datum in the report is the following graph:

It is accompanied by the following text:A second issue raised by this data is the role of computation and the different picture of computation across the three grades. For example, at both sixth and seventh grades adding Computation to Math Total lowers the CMP gain scores while raising the Non-CMP gain scores. Furthermore, with Computation included, CMP gains statistically less at sixth grade, gains the same at seventh grade, and gains statistically more at eighth grade. What would account for these patterns? Figure 2 shows displays this data graphically. On the face of it, this graph seems to imply that there is a consistent improvement in the scores of the CMP students when compared to non-CMP students. One mathematician who has carefully read this paper had the following observations.Look at the graph called Figure 2 "Fall to Spring ITBS Scores". You will notice that the non-CMP students seem to get dumber as the years go by, and seem to forget more over the summers than they learn during the year. The reason is obvious: the non-CMP students were 3 separate groups of students, not the same set over three years, (On page 2, the paper suggests that this is the case). NO effort was made to calibrate differences. It is pretty clear what is happening: the grade 6 data is from a school where the honors kids were non-CMP and the grade 8 data is from a school where only remedial kids were non-CMP.They do not even try to disguise this.There is a second paper extolling CMP at the CMP website by Reys, et.al. In this paper the statistics are done well but the "control group" is not realistic. The paper looks at three programs: CMP, another similar program, and a "control group" that consists of teachers who seem to share the same philosophy as the developers of CMP but are teaching without the assistance of any books or course materials. In other words the control group consists of teachers who are just winging it.Unfortunately, this kind of statistical analysis, poorly done and misleading, appears to be very common in research on NSF funded programs, and the errors all seem to be in the direction most favorable to the programs. For example one can check Kim Mackey's analysis of similar research reports on CorePlus on the math-teach archive at Swarthmore: http://forum.swarthmore.edu/epigone/math-teachApril 11 - 14, 1999, Core-Plus Evaluation, Parts I - IV.Finally, the site contains a ringing endorsement from the AAAS. Here are the comments of one of the professors in the department of mathematics at Michigan State explaining the significance of this endorsement."I am a MSU Math professor. While I am a research mathematician, I have been teaching courses in Math Education, and have been closely following current developments in Math education. Evaluating Math programs is a tricky business in the current environment. There are major battles going on, and reports from even trusted sources are usually tinged with the politics of these battles.The AAAS report is a case in point. It was not written by scientists. Rather, the AAAS has lent its imprimatur, under the name `Project 2061' , to a group of EDUCATORS who are not trained in mathematics or science. These educators have a specific political agenda. They would like to see all education done in group settings with the `discovery method' and with no direct instruction from the teacher. They would like to see mathematics classes with long writing assignments, no right and wrong answers, no practice problems, complete reliance on calculators, and a minimization of algebra.Accordingly, they designed a set of criteria focusing not on WHAT and HOW mathematics is covered by the program, but rather on the extent to which it conforms to the above agenda. Take a close look at the Project 2061 websitehttp://project206 1.aaas.org/newsinfo/press/attach_a.pdfand you will quickly see that this is so."EmailDouble click on thumbnails to get full-size. I recommend that you copy the pictures and information you find valuable. To those of you who have contributed to our body of knowledge on these families, I thank you. You know who you are and if by some chance, you are not listed on these pages, it is not an intentional error, merely something that happens on uploading the files from my genealogy program to the web page from which everyone will benefit. I do not claim to have done all the research nor do I claim to have authority for what is included. It looks to me the best information to me and you should make the effort to verify. I have attempted to give credit within the notes. Sharman Ramsey